Common divisors survive mixing
A common divisor of two numbers automatically divides every 'linear combination' you can build from them. This tiny fact is a workhorse: it means the set of numbers d divides is closed under the most common algebraic moves, so divisibility facts propagate through calculations.
If two paints are both mixed from the same base pigment, any blend of them still contains that pigment.
3 divides 12 and 21. Check 12×5 − 21×2 = 60 − 42 = 18 — and 3 | 18, as promised. It works for EVERY choice of the multipliers.
This is the step that makes the Euclidean algorithm work (gcd(a,b) = gcd(b, a−b)) and quietly appears inside nearly every divisibility proof.
Level 1 The precise statement
If d | a and d | b, then d | (a*x + b*y) for all integers x, y. In particular d | (a - b).
Level 2 The proof
Level 3 What it stands on (2 direct)
- Integers form a commutative ring (axiom)
- Divisibility (definition)
- Divisibility respects linear combinations (lemma)
Level 4 The verified record
This page is generated from a machine-checked node. The kernel confirms its dependencies resolve, nothing is circular, and it grounds in axioms (foundation: peano). The content hash below makes tampering evident.
2 downstream results would collapse with it. See the blast radius on the graph →